Mathematics – Differential Geometry
Scientific paper
2012-03-21
Mathematics
Differential Geometry
28 pages
Scientific paper
For the cotangent bundle $T^{*}K$ of a compact Lie group $K$, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space $L^{2}(K)$ under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between $L^{2}(K)$ and a certain weighted $L^{2}$-space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time $-\tau$) within $L^{2}(K)$, followed by a polarization--changing geometric quantization evolution (for complex time $+\tau$). In this case, our construction yields the usual generalized Segal--Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemann's "complexifier" method (which generalizes the construction of adapted complex structures). We will also investigate some properties of the generalized CSTs, and discuss how their existence can be understood in terms of Mackey's generalization of the Stone-von Neumann theorem.
Kirwin William D.
Mourao José M.
Nunes Joao P.
No associations
LandOfFree
Complex time evolution in geometric quantization and generalized coherent state transforms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Complex time evolution in geometric quantization and generalized coherent state transforms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Complex time evolution in geometric quantization and generalized coherent state transforms will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-488773